General properties of the Helmholtz decomposition in spaces of L~9-type

摘要

We discuss general properties of the Helmholtz projection and its relation to variational (in-)equalities for gradients and solenoidal function spaces. This discussion is extended to the case where the classical algebraic and topological Helmholtz decomposition fails, either since the decomposition is not unique or does not hold for the complete L~q-space. Moreover, we extend the Helmholtz decomposition to the case where it holds for some qo,q'_0 ≠ 2 but not for all q ≠ 2 by working in suitable intersection and sum spaces defined by L~(q0), L~(q'o) and L~q.